CIW Math refresher

Tina Rozsos

VU Amsterdam

October 15, 2025

Math and stats refresher

Overall goal

  • Revise the math & stats concepts you need for the Choices, Inequality, and Welfare course
  • Focus is on
    • intuitive understanding of concepts
    • ability to read and use math notation and graphical representations

Plan for today

  • solve and discuss exercises
  • questions from self-test, then other topics you want to discuss

These slides complement the math self-test on Desmos and the math review handout

Graphing functions

  1. Consider the function \(f(x) = 2x-1\)
  2. Goal: plot the values \(f(x)\) takes for different values of \(x\)
  3. Horizontal axis: \(x\), vertical axis: \(f(x)\)
  4. For example, for \(x=0\), \(f(0) = 2*0-1 = -1\)
  5. For \(x=1\), \(f(1) = 2*1-1 = 1\)
  6. Looking at the function, we can tell it’s linear – if \(x\) increases by 1, \(f(x)\) always increases by 2
  7. So we can take two points and draw a straight line through them

Graphing functions

Graphing functions

Graphing functions (quadratic)

  1. Example: \(f(x) = (x-2)^2+3\)
  2. Quadratic functions are drawn as parabolas
  3. To draw the function, find the vertex, and decide if it is a minimum or maximum
  4. Finding the vertex:
    1. given \(f(x) = a(x-h)^2+k\), the vertex is at \((h,k)\)
    2. or calculate the derivative and set it to 0:
      1. given \(f(x) = ax^2+bx+c\), \(f'(x)=2ax+b\)
      2. \(2ax+b=0\) gives the \(x\)-coordinate of the vertex
      3. use the original \(f(x)\) to find the \(y\)-coordinate
  5. The sign of the \(x^2\) term tells you if the parabola opens up (+) or down (-)
  6. Calculating some more points to each side of the vertex helps to draw the parabola, e.g. for \(x=0\), \(f(0) = (0-2)^2+3 = 7\)

Graphing functions (quadratic)

Convexity and concavity

  1. A function is convex if it lies on or below the line connecting any two points on the function
    • Analytical definition for twice-differentiable functions: \(f''(x) \ge 0\) for all \(x\)
  2. A function is concave if it lies on or above the line connecting any two points on the function
    • Analytical definition for twice-differentiable functions: \(f''(x) \le 0\) for all \(x\)
  3. A function is strictly convex (concave) if it lies strictly below (above) the line connecting any two points on the function

Monotonicity

  1. A function is monotonically increasing if \(f(x_2) \ge f(x_1)\) whenever \(x_2 > x_1\)
  2. A function is monotonically decreasing if \(f(x_2) \le f(x_1)\) whenever \(x_2 > x_1\)
  3. Strictly monotonic: inequalities are strict (\(>\) or \(<\))
  4. Monotonicity is about the direction of change, not curvature

Functions in economics

In economics, functions rarely use \(x\) and \(y\)

Instead: economic variables

E.g. price \(P\), quantity demand \(Q_D\)

  • Axes: price \(P\) on vertical axis, quantity \(Q_D\) on horizontal axis
    • rearrange equations for plotting:
    • \(P = 20 - \frac{1}{5} Q_D\)
    • \(P = -\frac{20}{3} + \frac{1}{3} Q_S\)
  • Movement along a curve: when \(P\) goes up, \(Q_D\) goes down
  • Shift of a curve: the relationship between \(P\) and \(Q_D\) changes
    • e.g. income \(Y\) increases \(\to\) demand curve shifts right; more demand at any price

Demand: \(Q_D = 100 - 5P\)

Supply: \(Q_S = 20 + 3P\)

Calculating areas

Areas are often needed to compute surplus or costs

  • Area of a triangle: \(\frac{1}{2} \times \text{base} \times \text{height}\)
  • Area of a trapezoid: \(\frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}\)

Example: consumer surplus under a linear demand curve

\(\text{Consumer Surplus} = \frac{1}{2} \times (50-0) \times (20-10) = 250\)

Logarithms

Elasticities

  • Elasticity tells how sensitive one variable is to another
  • E.g. “If price goes up 1%, quantity demanded goes down by 0.5%”
  • Unit-free, so we can compare across contexts (e.g. price vs income effects)

Optimization

Optimization = finding maximum or minimum of a function

  1. Set derivative to zero: \(f'(x)=0\)
  2. Second derivative test:
    • \(f''(x) > 0\) → minimum
    • \(f''(x) < 0\) → maximum

Example: profit maximization with \(\pi(q) = 10q - q^2\)

\(\pi'(q) = 10 - 2q = 0 \Rightarrow q^* = 5\)

Distributions

  1. A distribution describes how likely different values of a variable are
  2. A probability distribution function (pdf) gives the probability of each value (in discrete case; in continuous case, the pdf is a density: area under the curve gives probabilities over intervals)
  3. The cumulative distribution function (cdf) gives the probability of a value being less than or equal to a certain value

Distributions

Consider the data:

(1, 1, 3, 4, 5, 5, 5, 6, 9, 10)

Plot the distribution of the data:

Distributions

Consider the data:

(1, 1, 3, 4, 5, 5, 5, 6, 9, 10)

Rescale the frequencies to get a probability distribution:

Distributions

Consider the data, assuming it comes from a continuous distribution (i.e. any value is possible):

(1, 1, 3, 4, 5, 5, 5, 6, 9, 10)

Smooth out the histogram to get a density plot:

Distributions

Consider the data, assuming it comes from a continuous distribution (i.e. any value is possible):

(1, 1, 3, 4, 5, 5, 5, 6, 9, 10)

The area of the density plot is 1, meaning that the probability of the variable taking any value within the full range is 1:

Distributions

At each value of \(x\), the CDF gives the area under the density curve to the left of \(x\).

Using the discrete example from above, adding up the area of the histogram bars up to \(x\) gives the height of the CDF at each value of \(x\):